# Measurements & Calculations

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Measurements & Calculations
Chapter 2

Scientific Notation Section 2.1
Objective: show how very large or very small numbers can be expressed as the product of a number between 1 and 10 AND a power of 10

(the number of atoms in a drop of water)
Scientific Numbers In science, we often encounter very large and very small numbers. Using scientific numbers makes working with these numbers easier 5,010,000,000,000,000,000,000 a very large number (the number of atoms in a drop of water) a very small number (mass of a gold atom in grams)

Scientific Numbers RULE 1 As the decimal is moved to the left The power of 10 increases one value for each decimal place moved Any number to the Zero power = 1 450,000,000 = 450,000,000. x 10 8 2 3 1 450,000,000 = 450,000,000. x 10 8 4.5 x 10

Scientific Numbers 0.0000072 = 0.0000072 x 10 -6 -2 -3 -1
RULE 2 As the decimal is moved to the right The power of 10 decreases one value for each decimal place moved Any number to the Zero power = 1 = x 10 -6 -2 -3 -1 = x 10 -6 7.2 x 10

UNITS and MEASUREMENTS OF LENGTH, VOLUME & MASS
Section 2.2 and 2.3 UNITS and MEASUREMENTS OF LENGTH, VOLUME & MASS Objective: to learn english, metric and SI systems of measurement

Measurements of length, volume & mass
Section 2.3 Measurements of length, volume & mass Objective: understand metric system for measuring length, volume and mass

Property English Unit Metric Unit
Fundamental Quantities Property English Unit Metric Unit mass slug gram 1.0 slug ,590 g length foot meter 1.0 ft m volume quart liter 1.06 qt L

Property Metric Unit English Unit
time second second temperature Kelvin Fahrenheit

Significant Figures Section 2.5
Objective: to learn how to determine the number of sig figs

Significant Figures All number other then zero are significant
Ex. 23 = 2 sig figs Leading zeros- zeros that are at the beginning of a number are NEVER significant Ex 034 = 2 sig figs and = 3 sig figs Trapped zeros – zeros that are trapped between two other significant figures are ALWAYS significant Ex 304 = 3 sig figs and = 5 sig figs Trailing zeros – zeros that are at the end of a number – depends on if there is a decimal point expressed in that number If there is a decimal point showing in the number then the zeros are significant Ex 60 = 1 si fig but 60. = 2 sig figs and 60.0 = 3 sig figs Ex .05 = 1 sig figs If there is NOT a decimal point showing in the number then the zeros are NOT sinificant

Example: 120000

Significant Figures 120000 120000. No decimal point Zeros are not
2 sig figs Zeros are not significant! Decimal Point All digits including zeros to the left of The decimal are significant. 6 sig figs

Significant Figures 1005 123.00 All figures are Significant 4 sig figs
Zeros between Non zeros are significant 123.00 All figures are Significant 5 sig figs Zero to the Right of the Decimal are significant

Zeros to the right of the decimal And to the right of non zero values
Significant Figures 3 sig figs Zeros to the right of The decimal with no Non zero values Before the decimal Are not significant 5 sig figs Zeros to the right of the decimal And to the right of non zero values Are significant

Significant Figures 1 in = 2.54cm 4 quarts = 1 gallon
Exact equivalences have an unlimited number of significant figures 1 in = 2.54cm Therefore in the statement 1 in = 2.54 cm, Neither the 1 nor the 2.54 limits the number of Sig figs when used in a calculation The same is true for: 4 quarts = 1 gallon 100 centimeters = 1 meter 1000 grams = 1 kilograms and so on !

(numbers that were not obtained using measuring
Exact numbers (numbers that were not obtained using measuring devices, but determined by counting) also have an unlimited number of sig figs Examples: 3 apples 8 molecules 32 students

Uncertainty in Measurement
Section 2.4 Uncertainty in Measurement Objective: to understand how uncertainty in measurement arises Difference between accuracy and precision

Significant Figures Significant figures are used to distinguish truly measured values from those simply resulting from calculation. Significant figures determine the precision of a measurement. Precision refers to the degree of subdivision of a measurement. As an example, suppose we were to ask you to measure how tall the school is, you replied “About one hundred meters”. This would be written as 100 with no decimal point included. This is shown with one significant figure the “1”, the zeros don’t count and it tells us that the building is about 100 meters but it could be 95 m or even 104 m. If we continued to inquire and ask you to be more precise, you might re-measure and say “ OK, ninety seven meters. This would be written as 97m. It contains two significant figures, the 9 and the 7. Now we know that you have somewhere between 96.5m and 97.4m. If we continue to ask you to measure even more precise with more precision, may eventually say, “97.2 m”. THE PRECISION OF YOUR MEASUREMENT IS DICTATED BY THE INSTRUMENT YOU ARE USING TO MEASURE!!!!

Precision = Accuracy ACCURACY MEANS HOW CLOSE A MEASUREMENT
IS TO THE TRUE VALUE PRECISION REFERS TO THE DEGREE OF SUBBDIVISION OF THE MEASUREMENT FOR EXAMPLE, IF A ROOM IS 10 FEET LONG AND YOU MEASURE IT TO BE FT LONG, YOUR MEASUREMENT IS VERY PRECISE BUT INACCURATE ! MEASUREMENTS SHOULD BE ACCURATE AND AS PRECISE AS THE MEASURING DEVICE ALLOWS

Illustration of accurate vs percise

You tell me. What is it?

Precision & Measurement
Measurements are always all measured values plus one approximated value. The pencil is 3.6 cm long. With more calibration a more precise measurement is possible The pencil is 3.64 cm long! 4 The calibration of the instrument determines measurement precision Now cm !

What is the precision on a ruler?
Follow directions from Mrs McGrath & try to figure it out??? What if your measurement was in cm? What if your measurement was in mm?

Scales and sig figs In our class Write down what the scale says
Most scales are taken to the hundreths place

First – figure out scale Then – take measurement out to one guess past certainity

Section 2.5 continued Math and Sig Figs

Rules for rounding off In calculations:
Round each number to one sig fig: If the digit is to be removed: Is less than 5, the preceding digit stays the same EX rounds to ???? _____________ is equal to or greater then 5, the preceding digit is increased EX rounds to ???? _____________ In calculations: carry the extra digits through to the final result AND THEN round off

Adding and subtracting with significant figures. The position, not the number, of the significant figures is important in adding and subtracting. For example, (the last sig fig is in hundredth place (0.01)) (the last sig fig is in ten thousandth (0.0001)) (the answer is rounded off to the least significant position hundredths place)

The answer is rounded to the position of least significance
Adding & Subtracting Sig Figs The numbers in these positions are not zeros, they are unknown 123.6 _ Don’t even look at The 6 to determine Rounding. Only Look at the 4 165.9 The answer is rounded to the position of least significance

Multiplying/Dividing with Significant Figures
The result of multiplication or division can have no more sig figs than the term with the least number. *ex. 9 x 2 = 20 since the 9 has one sig fig and the 2 has one sig fig, the answer 20 must have only one and is written without a decimal to show that fact. * By contrast, 9.0 x 2.0 = 18 each term has two sig figs and the answer must also have two. *4.56 x 1.4 = How many sig figs can this answer have? 6.4 (2 sig figs)

Dimensional analysis Section 2.6
Objective: learn how to apply dimensional analysis to solve problems

NO KING HENRY You must use dimensional analysis to convert from metric to metric You must use your brain and logic to do this K H D b d c m

Some Common Metric Prefixes
Prefix Multiplier Example milli milliliter centi centimeter deci decigram kilo kilometer micro microgram 6 Mega megabyte 9 Giga gigabyte

From the last slide we learned the meaning of some of the common prefixes, BUT we are going to learn to dimensional analysis using the root prefixes and deciding bigger/smaller.

Some Common Metric Prefixes
Prefix Multiplier Prefix milli milli centi centi deci deci kilo kilo micro micro 6 Mega mega 9 Giga giga

Conversions YOU Need to Memorize
Length 1in = 2.54 cm 39.37 in = 1 meter 1 mile = 5280 feet Mass 1kg = 2.2 lbs 1lb = 454 grams Volume 1 liter = 1.06qts 1 gallon = 3.79 liters

Dimensional Analysis Rules
1.37days = ? minutes Always start with the known value over the number 1 Always write one number over the other Always, Always, Always, Always, Always write/include the unit with the number 1.37 days 1

Single step examples Equivalence statements 3.6 m = ? ft 6.07 lb = ?kg
4.2 L = ?qt 35.92 cm = ? in Length 1in = 2.54 cm 39.37 in = 1 meter 1 mile = 5280 feet Mass 1kg = 2.2 lbs 1lb = 454 grams Volume 1 liter = 1.06qts 1 gallon = 3.79 liters

Double step Exampls Equivalence Statements 56,345 s = ? yrs
98.3 in = ?m 3.2 mi = ?km Length 1m = yd 2.54 cm = 1 in 1mi = 1760 yd Mass 1kg = lb 453.6 g = 1lb Volume 1 L = 1.06qt

Temperature Conversions
Section 2.7 Temperature Conversions Objective: to learn three temperature scales to convert from one scale to another

Temperature – the average kinetic energy in a substance
Boiling points Fahrenheit 212 F Celsius 100 C Kelvin 373 K Freezing points Fahrenheit 32 F Celsius 0 C Kelvin K *O Kelvin or Absolute zero: point at which molecular motion stops

Temperature Conversion Formulas
Celsius to Kelvin TK = TC + 273 Kelvin to Celsius TC = TK – 273 Celsius to TF = 1.80TC Fahrenheit Fahrenheit to TC = TF - 32 Celsius

Section 2.8 Density Objective: to define density and its units

Density: the amount of matter present in a given volume of a substance
Units Formula Density = g/ml OR g/cm3 Mass = g (grams) Volume = ml OR cm3 Liquids OR solids Density = mass/volume DENSITY of a substance never changes Ex gold is ALWAYS 19.3g/cm3 Less dense objects “FLOAT” in more dense objects

Example calculation Mercury has a density of 13.6g/ml. What volume of mercury must be taken to obtain 225 grams of the metal?